Thirring model
thirring modelThe Thirring model is an exactly solvable quantum field theory which describes the selfinteractions of a Dirac field in 1+1 dimensions
Contents
 1 Definition
 2 Massless case
 21 Exact solution
 3 Massive Thirring model, or MTM
 31 Exact solution
 4 Bosonization
 5 See also
 6 References
 7 External links
Definition
The Thirring model is given by the Lagrangian density
L = ψ ¯ i ∂ / − m ψ − g 2 ψ ¯ γ μ ψ ψ ¯ γ μ ψ }=}i\partial \!\!\!/m\psi }\left}\gamma ^\psi \right\left}\gamma _\psi \right\ }where ψ = ψ + , ψ − ,\psi _} is the field, g is the coupling constant, m is the mass, and γ μ } , for μ = 0 , 1 , are the twodimensional gamma matrices
This is the unique model of 1+1dimensional, Dirac fermions with a local selfinteraction Indeed, since there are only 4 independent fields, because of the Pauli principle, all the quartic, local interactions are equivalent; and all higher power, local interactions vanish Interactions containing derivatives, such as ψ ¯ ∂ / ψ 2 }\partial \!\!\!/\psi ^} , are not considered because they are nonrenormalizable
The correlation functions of the Thirring model massive or massless verify the OsterwalderSchrader axioms, and hence the theory makes sense as a quantum field theory
Massless case
The massless Thirring model is exactly solvable in the sense that a formula for the n points field correlation is known
Exact solution
After it was introduced by Walter Thirring, many authors tried to solve the massless case, with confusing outcomes The correct formula for the two and four point correlation was finally found by K Johnson; then C R Hagen and B Klaiber extended the explicit solution to any multipoint correlation function of the fields
Massive Thirring model, or MTM
The mass spectrum of the model and the scattering matrix was explicitly evaluated by Bethe Ansatz An explicit formula for the correlations is not known J I Cirac, P Maraner and J K Pachos applied the massive Thirring model to the description of optical lattices
Exact solution
In one space dimension and one time dimension the model can be solved by the Bethe Ansatz This helps one calculate exactly the mass spectrum and scattering matrix Calculation of the scattering matrix reproduces the results published earlier by Alexander Zamolodchikov The paper with the exact solution of Massive Thirring model by Bethe Ansatz was first published in Russian Ultraviolet renormalization was done in the frame of the Bethe Ansatz The fractional charge appears in the model during renormalization as a repulsion beyond the cutoff
Multiparticle production cancels on mass shell
The exact solution shows once again the equivalence of the Thirring model and the quantum sineGordon model The Thirring model is Sdual to the sineGordon model The fundamental fermions of the Thirring model correspond to the solitons of the sineGordon model
Bosonization
S Coleman discovered an equivalence between the Thirring and the sineGordon models Despite the fact that the latter is a pure boson model, massless Thirring fermions are equivalent to free bosons; besides massive fermions are equivalent to the sineGordon bosons This phenomenon is more general in two dimensions and is called bosonization
See also
 Dirac equation
 GrossNeveu model
 Nonlinear Dirac equation
 Soler model
References
 ^ Thirring, W 1958 "A Soluble Relativistic Field Theory" Annals of Physics 3: 91–112 Bibcode:1958AnPhy391T doi:101016/0003491658900150
 ^ Johnson, K 1961 "Solution of the Equations for the Green's Functions of a two Dimensional Relativistic Field Theory" Il Nuovo Cimento 20 4: 773 Bibcode:1961NCim20773J doi:101007/BF02731566
 ^ Hagen, C R 1967 "New Solutions of the Thirring Model" Il Nuovo Cimento B 51: 169 Bibcode:1967NCimB51169H doi:101007/BF02712329
 ^ Klaiber, B 1968 "The Thirring Model" Lect Theor Phys 10A: 141–176 OSTI 4825853
 ^ Cirac, J I; Maraner, P; Pachos, J K 2010 "Cold atom simulation of interacting relativistic quantum field theories" Physical Review Letters 105 2: 190403 arXiv:10062975 Bibcode:2010PhRvL105b0403B doi:101103/PhysRevLett105190403 PMID 21231152
 ^ Korepin, V E 1979 "Непосредственное вычисление Sматрицы в массивной модели Тирринга" Theoretical and Mathematical Physics 41: 169 Translated in Korepin, V E 1979 "Direct calculation of the S matrix in the massive Thirring model" Theoretical and Mathematical Physics 41 2: 953 Bibcode:1979TMP41953K doi:101007/BF01028501
 ^ Coleman, S 1975 "Quantum sineGordon equation as the massive Thirring model" Physical Review D 11 8: 2088 Bibcode:1975PhRvD112088C doi:101103/PhysRevD112088
External links
 On the equivalence between sineGordon Model and Thirring Model in the chirally broken phase





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